Skip to main content

From the Hartree Dynamics to the Vlasov Equation

Abstract

We consider the evolution of quasi-free states describing N fermions in the mean field limit, as governed by the nonlinear Hartree equation. In the limit of large N, we study the convergence towards the classical Vlasov equation. For a class of regular interaction potentials, we establish precise bounds on the 0rate of convergence.

This is a preview of subscription content, access via your institution.

References

  1. Amour L., Khodja M., Nourrigat J.: The semiclassical limit of the time dependent Hartree–Fock equation: the Weyl symbol of the solution. Anal. PDE 6(7), 1649–1674 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  2. Amour L., Khodja M., Nourrigat J.: The classical limit of the Heisenberg and time dependent Hartree–Fock equations: the Wick symbol of the solution. Math. Res. Lett. 20(1), 119–139 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  3. Athanassoulis A., Paul T., Pezzotti F., Pulvirenti M.: Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Rend. Lincei Mat. Appl. 22, 525–552 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Bach, V., Breteaux, S., Petrat, S., Pickl, P., Tzaneteas, T.: Kinetic energy estimates for the accuracy of the time-dependent Hartree–Fock approximation with Coulomb interaction (preprint). arXiv:1403.1488

  5. Bardos C., Golse F., Gottlieb A.D., Mauser N.J.: Mean field dynamics of fermions and the time-dependent Hartree–Fock equation. J. Math. Pures Appl. (9) 82(6), 665–683 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  6. Benedikter, N., Jaksic, V., Porta, M., Saffirio, C., Schlein, B.: Mean-field evolution of fermionic mixed states Comm. Pure Appl. Math. doi:10.1062/cpa.21538

  7. Benedikter N., Porta M., Schlein B.: Mean-field evolution of fermionic systems. Commun. Math. Phys. 331, 1087–1131 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. Dobrushin R.L.: Vlasov equations. Funct. Anal. Appl. 13(2), 115–123 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  9. Elgart A., Erdos L., Schlein B., Yau H.-T.: Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. (9) 83(10), 1241–1273 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  10. Fröhlich J., Knowles A.: A microscopic derivation of the time-dependent Hartree–Fock equation with Coulomb two-body interaction. J. Stat. Phys. 145(1), 23–50 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. Gasser I., Illner R., Markowich P.A., Schmeiser C.: Semiclassical, t → ∞ asymptotics and dispersive effects for HF systems. Math. Model. Numer. Anal. 32, 699–713 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Graffi S., Martinez A., Pulvirenti M.: Mean-field approximation of quantum systems and classical limit. Math. Models Methods Appl. Sci. 13(1), 59–73 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  13. Lions P.-L., Paul T.: Sur les mesures de Wigner. Rev. Mat. Iberoam. 9, 553–618 (1993)

    MathSciNet  Article  Google Scholar 

  14. Markowich P.A., Mauser N.J.: The classical limit of a self-consistent quantum Vlasov equation. Math. Models Methods Appl. Sci. 3(1), 109–124 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  15. Narnhofer H., Sewell G.L.: Vlasov hydrodynamics of a quantum mechanical model. Commun. Math. Phys. 79(1), 9–24 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  16. Prodan E., Nordlander P.: Hartree approximation I: The fixed point approach. J. Math. Phys. 42, 3390–3406 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. Pezzotti F., Pulvirenti M.: Mean-field limit and semiclassical expansion of a quantum particle system. Ann. Henri Poincaré 10(1), 145–187 (2009)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. Petrat S., Pickl P.: A new method and a new scaling for deriving fermionic mean-field dynamics. arXiv:1409.0480

  19. Prodan E.: Symmetry breaking in the self-consistent Kohn–Sham equations. J. Phys. A Math. Gen. 38, 5647–5657 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. Solovej J.P.; Many body quantum mechanics. In: Lecture Notes, Summer 2007. https://www.mathematik.uni-muenchen.de/~lerdos/WS08/QM/solovejnotes.pdf

  21. Spohn H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3(4), 445–455 (1981)

    ADS  MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chiara Saffirio.

Additional information

Communicated by S. Serfaty

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Benedikter, N., Porta, M., Saffirio, C. et al. From the Hartree Dynamics to the Vlasov Equation. Arch Rational Mech Anal 221, 273–334 (2016). https://doi.org/10.1007/s00205-015-0961-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0961-z

Keywords

  • Initial Data
  • Coherent State
  • Vlasov Equation
  • Slater Determinant
  • Trace Class Operator